[Psychology] Mechanical Assemblies Phần pot

Not only must the error angle between peg and hole be less than the allowed wobble angle, as shown in Figure 10-17, but the maximum angular error is also governed by the coefficient of f

FIGURE 10-15 Geometry of a Two-Point Contact.

The variable c is called the clearance ratio It is the

di-mensionless clearance between peg and hole Figure 10-16

shows that the clearance ratio describes different kinds of

parts rather well That is, knowing the name of the part

and its approximate size, one can predict the clearance

ratio with good accuracy The data in this figure are

de-rived from industry recommended practices and ASME

standard fit classes ([Baumeister and Marks])

Equation (10-2) shows that as the peg goes deeper into

the hole, angle 0 gets smaller and the peg becomes more

parallel to the axis of the hole This fact is reflected in the

long curved portion of Figure 10-12

Figure 10-17 plots the exact version of Equation (10-2)

for different values of clearance ratio c Note particularly

the very small values of 9 that apply to parts with small

values of c Intuitively we know that small 9 implies

dif-ficult assembly Combining Figure 10-17 with data such

as that in Figure 10-16 permits us to predict which kinds

of parts might present assembly difficulties

The dashed line in Figure 10-17 represents the fact that

there is a maximum value for 9 above which the peg cannot

even enter the hole This value is given by

10.C.4 Wedging and Jamming

Wedging and jamming are conditions that arise from theinterplay of forces between the parts To unify the discus-sion, we use the definitions in Figure 10-9, Figure 10-10,and Figure 10-18 The forces applied to the peg by the

compliances are represented by F x, Fz, and M at or about

the tip of the peg The forces applied to the peg by its

contact with the hole are represented by f\, fa, and the

friction forces normal to the contacted surfaces The

co-efficient of friction is JJL (In the case of one-point contact,

there is only one contact force and its associated frictionforce.) The analyses that follow assume that these forcesare in approximate static equilibrium This means in prac-tice that there is always some contact—either one point ortwo—-and that accelerations are negligible The analysesalso assume that the support for the peg can be described

as having a compliance center

FIGURE 10-16 Survey of Dimensioning tice for Rigid Parts This figure shows that for a

Prac-given type of part and a two-decade range in ameters, the clearance ratio varies by a decade or less, indicating that the clearance ratio can be well estimated simply by knowing the name of the part.

di-10.C PART MATING THEORY FOR ROUND PARTS WITH CLEARANCE AND CHAMFERS 271

FIGURE 10-17 Wobble Angle Versus Dimensionless Insertion Depth Parts with smaller clearance ratio are

limited to very small wobble angles during two-point tact, even for small insertion depths Since successful as- sembly requires alignment errors between peg and hole axes to be less than the wobble angle, and since smaller errors imply more difficult assembly, it is clear that assem- bly difficulty increases as clearance ratio (rather than clear- ance itself) decreases.

con-FIGURE 10-18 Forces and Moments on a Peg ported by a Lateral Stiffness and an Angular Stiff-

Sup-ness Left: The peg is in one-point contact in the hole.

Right: The peg is in two-point contact.

and

respectively These formulas are valid for 9 <$C tan ' (//).

A force-moment equilibrium analysis of the peg in

one-point contact shows that the angle of the peg with respect

to the hole's axis is given by

where

SQ and #o, the initial lateral and angular error between peg and hole, are defined in Figure 10-9, while L g, the distancefrom the tip of the peg to the mathematical support point,

is defined in Figure 10-10

We can now state the geometric conditions for stage 1,the successful entry of the peg into the hole and the avoid-ance of wedging, in terms of the initial lateral and angularerrors To cross the chamfer and enter the hole, we need

10.C.4.a Wedging

Wedging can occur if two-point contact occurs when the

peg is not very far into the hole A wedged peg and hole

are shown in Figure 10-19 The contact forces f\ and /2

are pointing directly toward the opposite contact point and

thus directly at each other, creating a compressive force

inside the peg The largest value of insertion depth I and

angle 9 for which this can occur are given by

FIGURE 10-19 Geometry of Wedging Condition Left: The peg is shown with the smallest 9 and largest i for which

wedg-ing can occur, namely I = i^d The shaded regions, encloswedg-ing angle 20, are the friction cones for the two contact forces The

contact force can be anywhere inside this cone The two contact forces are able to point directly toward the opposite contact point and thus directly at each other This creates a compressive force inside the peg and sets up the wedge This can happen

only if each friction cone contains the opposite contact point Right: Once t > /j,d, this can no longer happen Contact force f-\

is at the lower limit of its friction cone while f-2 is at the upper limit of its cone, so that they cannot point right at each other.

where W is the sum of chamfer widths on the peg and

hole, and

If parts become wedged, there is generally no cure (if

we wish to avoid potentially damaging the parts) except to

withdraw the peg and try again It is best to avoid wedging

in the first place The conditions for achieving this,

Equa-tion (10-8) and EquaEqua-tion (10-9), can be plotted together as

in Figure 10-20 This figure shows that avoiding wedging

is related to success in initial entry and that both are

gov-erned by control of the initial lateral and angular errors

We can see from the figure that the amount of permitted

lateral error depends on the amount of angular error and

vice versa For example, we can tolerate more angular

er-ror to the right when there is lateral erer-ror to the left because

this combination tends to reduce the angular error during

chamfer crossing Since we cannot plan to have such

op-timistic combinations occur, however, the extra tolerance

does us no good, and in fact we must plan for the more

pessimistic case This forces us to consider the smallest

error window

Note particularly what happens if L g = 0 In this case

the parallelogram in Figure 10-20 becomes a rectangle and

all interaction between lateral and angular errors

disap-pears The reason for this is discussed above in connection

with Figure 10-14 This makes planning of an assembly

the easiest and makes the error window the largest

FIGURE 10-20 Geometry Constraints on Allowed Lateral and Angular Error To Permit Chamfer Crossing and Avoid

Wedging Bigger W, c, and e, and smaller \JL make the

par-allelogram bigger, making wedging easier to avoid Not only must the error angle between peg and hole be less than the allowed wobble angle, as shown in Figure 10-17, but the maximum angular error is also governed by the coefficient of

friction if wedging is to be avoided If L g is not zero, then

if there is also some initial lateral error, this error could be converted to angular error after chamfer crossing So, avoid- ing wedging places conditions on both initial lateral error and initial angular error The interaction between these con-

ditions disappears if L g = 0 This fact is shown intuitively in

Figure 10-14.

10.C.4.b JammingJamming can occur because the wrong combination ofapplied forces is acting on the peg Figure 10-21 states

that any combinations of the applied forces F x , F z , and M

which lie inside the parallelogram guarantee avoidance

10.C PART MATING THEORY FOR ROUND PARTS WITH CLEARANCE AND CHAMFERS 273

of jamming The equations that underlie this figure are

derived in Section 10.J.4 To understand this figure, it is

important to see the effect of the variable A This variable

is the dimensionless insertion depth and is given by

As insertion proceeds, both t and X get bigger This in

turn makes the parallelogram in Figure 10-21 get taller,

expanding the region of successful assembly The region

is smallest when A is smallest, near the beginning of

as-sembly We may conclude that jamming is most likely

when the region is smallest (Since the vertical sides of

the region are governed by the coefficient of friction /i,

the parallelogram does not change width during insertion

as long as /z is constant.)

If we analyze the forces shown on the right side of

Figure 10-18 to determine what F x, Fz, and M are for the

case where KQ is small, we find that

Fx = — F arising from deformation of Kx

M = LgF = -LgFx

Dividing both sides by rF z yields

(10-lla)

which says that the combined forces and moments on the

peg F x/ Fz andM/rFz must lie on a line of slope— ( L g / r ) passing through the origin in Figure 10-21 If L g /r is big, this line will be steep and the chances of F X /F Z and

M/rF z falling inside the parallelogram will be small

Sim-ilarly, if M/rF z and F X /F Z are large, the combination ofthese two quantities will define a point on the line that

is far from the origin and thus likely to lie outside theparallelogram

On the other hand, if L g/r is small so that the line is

about parallel to the sloping sides of the parallelogramwhen A is small, then the chance of the applied forcesfalling inside the parallelogram will be as large as pos-sible and will only increase as A increases Similarly if

M/rFz and F X/FZ are small, they will define a point onthe line that is close to the origin and thus be likely to lieinside the parallelogram When A is small and jamming

is most likely, the slope of sides of the parallelogram is

approximately /z Thus, if L g/r is approximately equal

to JJL, then the line, and thus applied forces and moments,

have the best chance to lie inside the parallelogram Since

JJL is typically 0.1 to 0.3, we see that the compliance center

should be quite near, but just inside, the end of the peg toavoid jamming

Instead of considering a single lateral spring ing the peg at the compliance center, let us imaginethat we have attached a string to the peg at this point

support-FIGURE 10-21 The Jamming Diagram This

dia-gram shows what combinations of applied forces and

moments on the peg F x / F z and M/r F z will permit sembly without jamming These combinations are rep- resented by points that lie inside or on the boundary

as-of the parallelogram A is the dimensionless insertion depth given in Equation (10-10) When A is small, in- sertion is just beginning, and the parallelogram is very small, making jamming hard to avoid As insertion pro- ceeds and A gets bigger, the parallelogram expands

as its upper left corner moves vertically upward and its lower right corner moves vertically downward As the parallelogram expands, jamming becomes easier

to avoid.

FIGURE 10-22 Peg in Two-Point Contact Pulled by Vector F This

models pulling the peg from the compliance center by means of a string.

See Figure 10-22 This again represents a pure force F

acting on the peg In this case, F can be separated into

components along F x and F z to yield

(10-12)

so that

(10-13)

which is similar to Equation (10-11) In this case, we can

aim the string anywhere we want but we cannot

indepen-dently set F x and F z But, by aiming the force, which

means choosing 0, we can make F x as small as we want,

forcing the peg into the hole As L g —>• 0, we can aim </>

increasingly away from the axis of the hole and still make

M and Fx both very small

In Chapter 9, a particular type of compliant support

called a Remote Center Compliance, or RCC, is described

which succeeds in placing a compliance center outside

it-self The compliance center is far enough away that there

is space to put a gripper and workpiece between the RCC

and the compliance center, allowing the compliance

cen-ter to be at or near the tip of the peg Thus L g —>• 0 if an

RCC is used

Figure 10-23 shows the configuration of the peg, the

hole, and the supporting stiffnesses when L g = 0 In this

case, K x hardly deforms at all This removes the source

of a large lateral force on the peg that would have acted

at distance L g from the tip of the peg, exerting a

con-siderable moment and giving rise to large contact forces

during two-point contact The product of these contact

FIGURE 10-23 When L g is Almost Zero, the Lateral Support Spring Hardly De- forms Under Angular Er- ror Compare the deformation

of the springs with that in ure 10-13, which shows the

Fig-case where L a » 0.

forces with friction coefficient /z is the main source ofinsertion force Drastically reducing these contact forcesconsequently drastically reduces the insertion force for agiven lateral and angular error Section 10.J derives allthese forces and presents a short computer program thatpermits study of different part mating conditions by cal-culating insertion forces and deflections as functions ofinsertion depth The next section shows example experi-mental data and compares them with these equations

10.C.5 Typical Insertion Force Histories

We can get an idea of the meaning of the above relations

by looking at a few insertion force histories These wereobtained by mounting a peg and hole on a milling machineand lowering the quill to insert the peg into the hole A6-axis force-torque sensor recorded the forces The pegwas held by an RCC The experimental conditions aregiven in Table 10-1

TABLE 10-1 Experimental Conditions for Part Mating Experiments

Support: Draper Laboratory Remote Center Compliance

Lateral stiffness = K x = 1 N/mm (40 Ib/in.)

Angular stiffness = K® = 53,000 N-mm/rad (470 in.-lb/rad)

Peg and hole: Steel, hardened and ground Hole diameter = 12.705 mm (0.5002 in.) Peg diameter = 12.672 mm (0.4989 in.) Clearance ratio = 0.0026

Coefficient of friction = 0.1 (determined empirically from one-point contact data)

M = -FxLg

10.C PART MATING THEORY FOR ROUND PARTS WITH CLEARANCE AND CHAMFERS 275

FIGURE 10-24 Insertion Force History The compliance

center is 4r back inside the peg from the tip There is lateral

error only, no angular error As expected, two-point contact

occurs, giving rise to the peak in the insertion force at a depth

of about 18 mm The peak at around 0 mm is due to chamfer

crossing Also shown on the plot is a theoretical estimate of

insertion force based on equations given in the Section 10.J.

A computer program in Section 10.J was used to create the

theoretical plot.

Figure 10-24 shows a typical history of F z for a case

where there is only lateral error and the compliance center

is about 4r away from the tip of the peg The first peak in

the force indicates chamfer crossing Between t — \ mm

and 1 = 9 mm is one-point contact, following which

two-point contact occurs The maximum force occurs at about

£ = 1 8 mm or about twice the depth at which two-point

contact began For many cases, we can prove that the peak

force will occur at this depth A sketch of the proof is in

Section 10.J

Figure 10-25 shows the insertion force for the case

where the lateral error is larger than that in Figure 10-24,

but L g is almost zero Here, there is essentially no

two-point contact, as predicted intuitively by Figure 10-14 and

Figure 10-23 Also shown is the lateral force F x These

results show the merit of placing the compliance center

near the tip of the peg

FIGURE 10-25 Insertion and Lateral Force History The

peg, hole, and compliant support are the same as in

Fig-ure 10-24, but L g is essentially zero As predicted, point contact does not occur, even though there is initially more lateral error than in Figure 10-24 This additional lat- eral error also is responsible for the larger chamfer crossing

two-force (the large spike at t = 0) in this case compared to

in an assembly workstation and describes how to calculatethe width of chamfers needed

While all of the figures in this chapter show chamfers

on the hole, the same conclusions can be drawn if thechamfer is on the peg If both peg and hole have chamfers,

then W in Equation (10-7) and Figure 10-20 is the sum of

the widths of these chamfers

Also, it is significant that if a properly designed pliant support is used, with its compliance center at the tip

com-FIGURE 10-26 Pictorial Summary of Conditions for

Successful Assembly of Round Pegs and Holes with

Chamfers.

of the peg, there will be little insertion force except thatgenerated by chamfer crossing As Chapter 11 shows, themagnitude of this force depends heavily on the slope andshape of the chamfers

While most chamfers are flat 45-degree bevels, somesolutions to rigid part mating problems have been based onchamfers of other shapes Figure 10-27 shows two exam-ples of designs for the ends of plug gauges Plug gauges aremeasuring tools used to determine if a hole is the correctdiameter To make this determination accurately requiresthat the clearance between hole and gauge be very small,making it difficult and time-consuming to insert and re-move the gauge, and to avoid wedging it in the hole Thedesigns in Figure 10-27 specifically prevent wedging bymaking the ends of the gauges spheres whose radii areequal to the peg's diameter The small undercut in thesecond design also helps to avoid damaging the rim ofthe hole

FIGURE 10-27 Two signs of Chamfer That Prevent Wedging Note that

De-the radius of De-the arc forming the nose of the peg is equal

in length to the diameter of the peg In order to avoid wedging, it is necessary to pivot the peg about the point where the nose becomes tangent to the straight side,

as shown at the right.

10.D CHAMFERLESS ASSEMBLY

Chamferless assembly is a rare event compared to

cham-fered insertion because only a few parts have to be made

without chamfers Many of these are parts of hydraulic

valves, whose sharp edges are essential for obtaining the

correct fluid flow patterns inside the valves In other cases,

chamfers must be very small due to lack of space; a

cham-fer always adds length to a part, and sometimes there is a

severe length constraint, either on a part or on the whole

product Chamferless assemblies are, of course, more

dif-ficult than chamfered ones because W in Equation (10-8)

is essentially zero An attempt to assemble such parts bydirectly controlling the lateral error to be less than theclearance is almost certain to fail This is especially true

of hydraulic valve parts, whose clearances are only 10 or

20 fim (0.0004" to 0.0008").

In spite of their relative rarity, chamferless assemblieshave attracted much research interest and some solutionsthat require active control, such as that in Figure 10-28.This is a multiphase method in which the peg is lowereduntil it strikes the surface well to one side of the hole The

10.D CHAMFERLESS ASSEMBLY 277

FIGURE 10-28 A Chamferless Assembly Strategy:

(1) Approach, (2) Slide laterally, (3) Catch the Rim of

the Hole and Tilt, (4) Lower Peg into Hole.

lateral error may not be known exactly but the direction

toward the hole is known well enough for the method to

proceed The peg is then slid sideways toward the hole

It is held compliantly near the top so that when it passes

over the edge of the hole its tip catches the rim of the

hole and it starts to tip over A sensor detects this tilt and

lateral motion is stopped and reversed slightly Hopefully

this allows the tip to fall slightly into the hole The peg is

then lowered carefully Rocking and lowering are repeated

until the peg is in

An elaboration of this strategy is employed by the

Hi-Ti Hand ([Goto et al.]), a motorized fine motion device

invented by Hitachi, Ltd In this method, if the peg meets

resistance during the lowering phase, it is gently rocked

side to side in two perpendicular planes The limits of

this rocking are detected by sensors, and the top of the

peg is then positioned midway between the limits The

peg is then pushed down some more or until resistance is

again detected This push and rock procedure is repeated

as necessary until the peg is all the way in In the case of the

Hi-Ti Hand, mating time is typically 3 to 5 seconds This

method is good if the parts are delicate because it

specif-ically limits the insertion force For parts that can stand

a little contact force, however, it is far too slow Typical

assembly times for chamfered parts held by an RCC are

of the order of 0.2 seconds

Figure 10-29 shows an entirely passive chamferless

assembly method ([Gustavson, Selvage, and Whitney])

"Passive" means that it contains no sensors or motors

Figure 10-30 is a schematic of the apparatus itself It has

several novel features, including two centers of

compli-ance which operate one after the other The operation

be-gins with the peg deliberately tilted into an angular error

and as little lateral error as possible (Note that this is

the opposite of the initial conditions for the Hi-Ti Hand,

where initial angular error is zero and there is deliberate

lateral error.) When the peg is tilted, one side of the peg

FIGURE 10-29 Passive Chamferless Assembly Strategy The inserter works by first permitting the peg to approach the hole tilted and then to turn up to an upright orientation with one edge slightly in the mouth of the hole Insertion proceeds from that point with the aid of a conventional RCC The de- tails of how this is accomplished are shown in Figure 10-30.

FIGURE 10-30 Schematic of Passive Chamferless

In-serter Left: Arrangement of the device while the peg is

ap-proaching the hole The first compliance center is active and the part can rotate around it because of the sprung linkage attached to the gripper The linkage is designed so that the tip of the peg does not move laterally very much while the peg is rotating up to vertical What little tip motion there is will be in a direction away from the first compliance center

so as to keep the tip pressed firmly against the rim of the hole By this means the peg is most likely to remain in the

mouth of the hole Right: The part has engaged the mouth of

the hole and is now locked into the vertical position Insertion proceeds from here the same as if there had been chamfers and chamfer crossing were complete.

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effectively acts as a chamfer, and it is almost certain that

the tip of the peg and mouth of the hole will meet Once

they meet, the gripper continues moving down while the

peg tilts up to approximately vertical under the influence

of the linkage which creates the first compliance center

Upon reaching vertical, the peg locks into the gripper and

comes under the influence of the compliant support above

10.E SCREW THREAD MATING

it, having the second center of compliance at the tip of thepeg The peg's tip stays in the mouth of the hole whilerotating up to vertical Insertion then proceeds as if theparts had chamfers, starting from the point where chamfercrossing is complete

Examples of the apparatus in Figure 10-30 are in useinstalling valves into automobile engine cylinder heads

Figure 10-4 showed normally mated screws Assembling

screws involves a chamfer mate similar to peg-hole

mat-ing followed by thread engagement The screw (or nut) is

then turned several turns until it starts to tighten The last

stage comprises tightening a specified amount

Aside from missing the mouth of the hole, screw

mat-ing can fail in two possible ways One is a mismatch of

threads caused by angular error normal to the insertion

direction The other is a mismatch caused by having the

peaks of the screw miss the valleys of the hole due to

an-gular error along the insertion direction Both of these are

interchangeably called "cross-threading."

In order for the threads to mismate angularly normal to

the insertion direction, the angular error must be greater

than the angle a between successive peaks or valleys,

de-fined in Figure 10-31

If we define the angle between peaks as a, the diameter

of the screw as d, and the thread pitch as p threads per

unit length, then

Values for a for different standard screw thread sizes

are shown in Figure 10-32 They indicate that for very

small screws, an angular error of 1.14 mrad or 0.8 degree

is enough to cause a tilt mismatch Angular control at this

level is comparable to that required to mate precision pegs

and holes, as indicated in Figure 10-17 For larger screws,

the angles become comfortably large, indicating what is

FIGURE 10-31 Schematic of Screw Thread Defining p and

d In order for threads to mismate

due to tilt angle error, the tilt must

be greater than a.

FIGURE 10-32 Maximum Permissible Angular Error sus Screw Size for UNC Threads to Prevent Tilt Mismatch Between Threads Since angular errors are relatively easy

Ver-to keep below a few tenths of a degree, angular threading is fairly easy to avoid for all but the smallest screws.

cross-found in practice, namely that this kind of error does nothappen very often since angular control as good as a de-gree or so is easy to obtain, even from simple tools andfixtures

The other kind of screw mating error is illustrated inFigure 10-33 Here, the error is also angular, but the angle

in question is about the insertion axis in the twist tion That is, the thread helices are out of phase Unlessthe materials of either the screw or the hole are soft, thiskind of error is also difficult to create

direc-Some study of this problem may be found inRussian papers Figure 10-34 and Figure 10-35 are from

[Romanov] The screw has a taper or chamfer of angle oc while the hole thread has a taper of angle y The analysis

in this paper is entirely geometric, with no consideration

of friction The conclusion is that a should be greater than

y (see Figure 10-36) This is an interesting conclusion

because the Russian standards at the time the paper was

written were a = 45 degrees, y = 60.

10.E SCREW THREAD MATING 279

FIGURE 10-33 Mismated Screws Due to Helical Phase Error The helices of the screw's

threads and the hole's threads are out of phase and have inter- fered plastically with each other.

FIGURE 10-34 Variables Involved in Predicting Screw

Cross-Threading ([Romanov])

Region 1: Adjacent Threads Crossed

Region 2: Screw Tilted ~ p/d

Region 3: Screw Tilted ~ 2p/d

Note: The graph is drawn for p/d = 0.156,

but graphs for other p/d are

Another method of aiding the starting of screws is todrastically change the shape of the tip Two examples areshown in Figure 10-37 These are called "dog point" and

"cone point" screws Each has two disadvantages—extracost and extra length—but the advantages are valuable.The dog point is a short cylinder that assures that the screw

is centered in the hole and parallel to it The cone pointprovides the largest possible chamfer, making it easier toput the screw in a poorly toleranced or uncertainly locatedhole, such as in sheet metal

The above methods of assembling screws all depend onthe helices mating with the correct phase without doinganything explicit to ensure that correct phase is achieved

A method that searches for the correct phase is the "turnbackwards first" method, known to work well with lids ofpeanut butter jars Usually this method requires sensing

To utilize it, one places screw and hole mouth-to-mouthand turns the screw backwards until one senses that the ithas advanced suddenly The magnitude of this advance isapproximately one thread pitch At this point, the threadsare in a dangerous configuration, with chamfered peaks al-most exactly facing each other So it is necessary to turn anadditional amount back, perhaps 45 degrees Then it is safe

to begin turning forwards If a full turn is made without

an advance being detected, successful mating will not bepossible, and the parts should be separated This method isslow and, as stated, requires sensing, but it works well andmay be necessary in the case of unusually large diametersand small thread pitches, where even small angular errorscan cause mismating

FIGURE 10-37 (a) Dog Point and (b) Cone Point Screws.

The last phase of screw mating is the tightening phase.

Screw tightening must be done with care in order to obtain

a properly and safely secured joint without risking

strip-ping the threads A commonly used but unreliable method

is to measure the torque required to tighten the screw The

unreliability is based on the fact that the felt torque is a

combination of tightening torque and friction torque

be-tween the head of the screw and the hole face Because of

the extra friction torque, one typically feels more torque

than is actually being exerted on the threads Errors of

50% or more are not unusual

A more reliable method measures both turn angle and

torque and seeks to set a certain amount of elongation into

the screw rather than to achieve a certain amount of torque

To achieve this, it is necessary to sense torque versus turn

angle and try to determine the inflection point of the curve

This point is related to the point at which the screw starts to

deform plastically, at which it has achieved its maximum

safe stretch For many screws, the entire tightening event

occurs within 1 to 10 degrees of rotation, as indicated in

Figure 10-38 Since screws are typically turned rapidly

un-of mismating has probably occurred.

by automatic screwdrivers, the measuring apparatus andbrakes on the screwdriver must act quickly Commercialdevices are available that operate on this principle A study

of torque-angle-controlled tightening of precision threads

by automatic control is given in [Dunne]

The last topic in this chapter is the assembly of gears This

is a complex topic on which only a little research has been

done We will assume that one gear has already been

in-stalled, and it is necessary to install and mate another or

others to it There are several cases to consider In each

case the common element is that gear mating requires two

separate alignments to occur One is to bring the pitch

cir-cles into tangency, and the other is to fit the teeth together

These two steps can be done in either order,

depend-ing on the circumstances Pitch circles are illustrated in

Figure 10-5

The first case analyzed is the easiest There is plenty

of space near the insertion point so the arriving gear may

be brought down to one side of its mate as shown in

Fig-ure 10-39 Once it is near, the tool rotates the gear about

its spin axis while bringing it laterally toward its mate

The mating direction is perpendicular to the spin axis of

the gears Eventually the teeth mesh and assembly can

continue So this method mates the teeth first and then the

pitch circles

If the arriving gear is on a shaft that must be inserted

into a bearing, the above method works if the teeth can

be mated before shaft and bearing If shaft and bearing

must mate first, then the best method is to spin the shaftand gear while inserting along the spin axis, in the hope

of mating the teeth The same problem arises if two gearsthat are linked together must mate simultaneously with athird gear, as shown in Figure 10-40 Thus this methodapproximately mates the pitch circles first and then matesthe teeth

However, an approach along the spin axis may notsucceed as easily as one perpendicular to it Gears are

FIGURE 10-39 The Side-Approach Method of Mating Gears In step 1, the gear is placed next to the mating gear.

In steps 2 and 3, the gear is moved toward its mate and is simultaneously rotated in one direction or in oscillation, until the teeth mate.

10.F GEAR MATING 281

FIGURE 10-40 The Spin-Axis-Approach Method of

Mat-ing Gears This method is often needed with planetary gear

trains.

designed so that when they are mated, with the pitch

cir-cles tangent, there is little or no clearance between

adja-cent teeth When gears are inserted along the spin axis, the

pitch circles are typically already approximately tangent

This method therefore depends on the teeth mating under

conditions in which there is little or no clearance between

them The arriving gear may simply come to rest on top

of its mate and spin without mating, especially if the pitch

circles overlap slightly

People typically make such mates by either (a)

wait-ing until a random chance mates the gears or (b) rockwait-ing

the arriving gear, tilting its spin axis away from parallel

to its final orientation, in order eventually to tilt the tip

of a tooth into the space between two teeth on the other

gear These random and unpredictable methods cannot be

used by automatic machinery without their being equipped

with extra degrees of freedom and sensors The method

also fails to have a predictable completion time, making it

an awkward one to include in an otherwise well planned

and rhythmic production line In short, the method lacks

structure and should be replaced with a better one

Two solutions are possible The first is shown in

Fig-ure 10-41 Here, a bevel has been cut on one side of the

teeth so that when they meet, the touching places will

not be on the pitch circle but instead somewhere else;

anywhere else will have larger clearance between mating

teeth, so the chance of mating will be much larger

The second solution is shown in Figure 10-42 This

idea is similar in spirit to the dog point screw To make it

work well, the chamfered pilot on the gear must be well

made so that it fits snugly within the teeth of the mating

gear This fit places the pitch circles close to each other

Spinning the arriving gear usually causes the teeth to mate

easily

FIGURE 10-41 Bevelling Gear Teeth to Aid Mating.

This idea is embodied in U.S Patent 4,727,770, which

is illustrated in Figure 10-43

Both of these solutions to gear mating have the samedisadvantage as dog point screws: They add length to thegears Since the length of a gear tooth's face is carefullycalculated to give the gear adequate load capacity and life,one does not shorten the face in order to accommodate ei-ther the bevel or the pilot Instead, one lengthens the gear

to provide space for the bevel or pilot while keeping thetooth face the same size An entire product can becomelonger if length is added to some of its parts, and the addedlength can be a problem for other reasons

The mating of splines is physically similar to mating

of gears Splines are essentially internal gear mates inwhich all the teeth mate at once since the pitch circles areconcentric

FIGURE 10-42 The "Dog-Point" Gear.

FIGURE 10-43 Richard Ordo's Patent on Multiple Gear Mating This patent uses the dog-point gear approach The

patented feature is inside the ellipse The mating situation shown here is similar to that shown in Figure 10-40.

10.G CHAPTER SUMMARY

This chapter has outlined the behavior of compliantly

supported rigid parts during the fine motion phase of

as-sembly The success of mating was shown to depend

heav-ily on the shapes of the parts, the initial errors between

them, the friction coefficient, and the compliance of the

supporting tools and grippers Success for chamfered and

chamferless peg-hole mates depends on avoiding

wedg-ing and jammwedg-ing The mathematical conditions for this

are shown and derived in Section 10.J All of the

rele-vant analyses assume that the parts are moving slowly

Conditions given for successful mating of screws andgears are geometrical since the theory is not well enoughdeveloped to provide anything else However, the condi-tions for successful assembly of simple peg-hole matesthat take account of friction are more restrictive than thepurely geometrical conditions That is, the allowed errorsare much smaller So it is likely that the geometric condi-tions given for gears and screws are also merely necessaryones and are not sufficient, implying that the true condi-tions are more restrictive

10.H PROBLEMS AND THOUGHT QUESTIONS

2 Using a micrometer or other appropriate measuring ments, measure the peg and hole diameters of several part mates

instru-in different products and accumulate statistics on percent of mates with clearance ratios in the ranges 0.0001 to 0.001, 0.001 to 0.01, and so on Compare your results to those shown in Figure 10-16 and Figure 10-17.

3 Obtain some close-fitting parts, such as a ball bearing and its housing, which have a clearance ratio in the range 0.001 to 0.003, approximately Verify first that the fit has clearance and that it

is possible to mate the parts without using force Then clean the parts thoroughly with soap and water Next, attempt to wedge the

1 Take apart a mechanical item (the stapler, a pump, toaster,

light fixture, etc.) and classify the part mates as follows:

Type of mate—peg/hole, press, tab/slot, screw, solder or

glue, thermal shrink, bayonet, compliant snap or wedge,

chamferless, and so on.

Direction of approach of mating parts with respect to each

other, based on a common coordinate frame attached to any

main part of your choice.

Accumulate the results for several products and create

statis-tics showing such things as percent occurrence of each mate

type and mate direction.

10.H PROBLEMS AND THOUGHT QUESTIONS

parts by pressing on the bearing on one side while it is part way into

the hole Finally, lubricate the parts and try again to wedge them.

Record your observations and explain them in terms of wedging

theory.

4 Explain in your own words the difference between wedging

and jamming.

5 Derive equations Equation (10-5), Equation (10-6), and

Equa-tion (10-7) Show all the necessary steps.

6 Derive Equation (10-11) Show all the necessary steps.

7 Explain carefully all the mating conditions all the way around

the periphery of the parallelogram in Figure 10-21 Include the

dotted vertical lines as well as the solid sloping lines and the four

heavy dots at the corners.

8 Prove the claim made in Section 10.C.4.b that the slope of

the sloping sides of the parallelogram in Figure 10-21 is

approxi-mately jU, when A is small.

9 Derive the coordinates of each of the four heavy dots in

Fig-ure 10-21—for example, the dot at (l//x, —1) Similarly, derive

the four intersections between the sloping lines and the graph

axes—for example, the intersection at (0, A.).

10 Draw a picture to show why the shaped ends of the pegs in

Figure 10-27 will not wedge.

11 Derive Equation (10-21) and Equation (10-22) Show all the

necessary steps.

12 Derive Equation (10-24) and Equation (10-25) Show all the

necessary steps.

13 Derive Equation (10-45) Show all the necessary steps.

14 Note that the part mating equations in Section 10.J have

been derived for the case where the initial errors #o and £Q are

both positive (In fact, the computer program listing in Table 10-2

in Section 10.J is valid only for this case and may give

meaning-less results or error bombs if other cases are tried.) This

repre-sents one of four possible cases, the others given by both errors

being negative or one of each being positive while the other is

negative Rederive equations Equation (10-21), Equation (10-22),

Equation (10-24), Equation (10-25), and Equation (10-45) for each

of the other three cases.

15 Note that the part mating equations in Section 10.J have

been derived for the case where the peg approaches the hole along

the hole axis Rederive the equations for the case where the peg

approaches along its own axis You will have to take care when

defining the initial errors, since the definitions used in the chapter may not be appropriate.

16 In Figure 10-44 is a sketch of a window sash The frame

squeezes the sash with equal friction force on both sides There is

a little side-to-side clearance between the sash and the frame To open this window most easily, should you push at A, B, C, D, E,

or F? Explain with words or equations as you prefer Ignore the mass of the sash.

FIGURE 10-44 A Window Sash.

The sash moves up and down in the frame.

17 In Figure 10-44, assume the friction force is bigger on the

left side than on the right To open the window most easily, where should you push? Explain with words or equations as you prefer.

18 In Figure 10-45 is a rod supported by a linear spring with

stiffness K K and an angular spring with stiffness Kg Write an pression for the total lateral displacement x-i that relates F, L g , and

ex-the two stiffnesses [The answer is provided here, but don't peek— use it only to check that you understand the problem.] Also, write

an expression for the angle 0.

FIGURE 10-45 A Rod Supported by a Linear Spring and

an Angular Spring.

Answer to Problem 18:

19 Continue with Problem 18 as follows:

a Show on Figure 10-45 where the compliance center is.

283

b Explain intuitively and with the aid of the equations how

X2 and 6 will behave if K x is zero or infinity and a force F

is applied as shown in Figure 10-45.

c Similarly, explain how X2 and 8 will behave if Kg is zero

or infinity.

d Finally, explain how X2 and 6 will behave if L g is zero.

20 Dan is frugal and brings home from business trips some

par-tially used little bottles of shampoo from hotel rooms He salvages

the shampoo by turning the little bottle up side down and carefully

placing its neck in the neck of a large bottle One time he arranged

them as shown in Figure 10-46.

He came back a while later to find the bottles as shown in

Figure 10-47 No jostling or vibration occurred to cause this Use

Figure 10-21 and Figure 10-22 as guides to explain what probably

FIGURE 10-48 Side and Front Views of a Luggage Bin Door on an Airplane.

FIGURE 10-46 A Little Shampoo tle Balanced on a Large One.

Bot Bot i

FIGURE 10-47 Configuration of the Two

Shampoo Bottles Later.

Detail of Door Opening

FIGURE 10-49 Detail of Latch When Door Is Closed.

When the handle is pulled to the left, the latch plate is posed to move up.

sup-4 While Dan was studying how the latch worked, another passenger leaned over and said, "Once an engineer, always an engineer!"

10.J APPENDIX 285

Dan got up and did a simple thing that permitted him to open

the door effortlessly Explain

To help you answer this question, look at Figure 10-48 and Figure 10-49, which diagram the door, how it is hinged, and how the latch works.

10.1 FURTHER READING

[Arai and Kinoshita] Arai, T, and Kinoshita, N., "The Part

Mat-ing Forces That Arise When UsMat-ing a Worktable with

Com-pliance," Assembly Automation, August, pp 204-210, 1981.

[Baumeister and Marks] Baumeister, T., and Marks, L S.,

Standard Handbook for Mechanical Engineers, 7th ed., New

York: McGraw-Hill, 1967.

[Dunne] Dunne, B J., "Precision Torque Control for Threaded

Part Assembly," M.S thesis, MIT Mechanical Engineering

Department, 1986.

[Gustavson, Selvage, and Whitney] Gustavson, R E., Selvage,

C C., and Whitney, D E., "Operator Member Erection

System and Method," U.S Patent 4,324,032, 1982.

[Goto et al.] Goto T., Takeyasu, K., and Inoyama, T.,

"Control Algorithm for Precision Insert Operation Robot,"

IEEE Transactions on Systems, Man, and Cybernetics, vol.

SMC-10, no 1, pp 19-25, 1980.

[Nevins and Whitney] Nevins, J L., and Whitney, D E., editors,

Concurrent Design of Products and Processes, New York:

McGraw-Hill, 1989.

[Romanov] Romanov, G I., "Preventing Thread Shear in

Au-tomatic Assembly," Russian Engineering Journal, vol 44,

no 9, pp 50-52, 1964.

[Simunovic] Simunovic, S., "Force Information in Assembly Processes," presented at the 5th International Symposium on Industrial Robots, Chicago, 1975.

[Sturges] Sturges, R H., Jr., "A Three-Dimensional Assembly Task Quantification with Application to Machine Dexterity,"

International Journal of Robotics Research, vol 7, no 4,

pp 34-78, 1988.

[Whitney] Whitney, D E., "Quasi-Static Assembly of

Compli-antly Supported Rigid Parts," Transactions of the ASME, Journal of Dynamic Systems, Measurement, and Control,

vol 104, pp 65-77,1982 This reference contains many other references to part mating theory.

10.J APPENDIX: Derivation of Part Mating Equations

This appendix sketches the derivations of the basic

equations for rigid part mating when the parts are

sup-ported by a support with a compliance center More

de-tail may be found in [Whitney] The derivations presume

that the compliance center is located on the peg's axis an

arbitrary distance L g from the tip of the peg Chamfer

crossing, one-point contact, and two-point contact will be

described The derived equations and computer program

treat the case where lateral error and angular error are both

positive as shown in Figure 10-9.

10.J.1 Chamfer Crossing

Refer to Figure 10-50, which shows a peg during chamfer

crossing and the forces on it.

The compliant support contributes the applied forces,

expressed as F x , F z , and M at the tip of the peg The

con-tact between peg and chamfer provides the reaction forces.

The support forces are found by determining how far the

compliances described by K x and KQ have been deflected.

The initial lateral displacement of the support point with

respect to the hole's axis is given by UQ:

When U = UQ and 0 = OQ, both compliances are laxed As chamfer crossing proceeds, U and 9 are related

re-by

why the door is hard to open

what Dan did and why it worked.

where a is defined in Figure 10-9 To find U and 9

separately, we have to solve for the forces and moments Writing equilibrium equations between the applied forces and contact forces yields

FIGURE 10-50 Left: Geometry of Chamfer Crossing Right: Forces During Chamfer Crossing.

where

and

Combining the above equations yields expressions for

U and 0 during chamfer crossing:

and

10.J.2 One-Point Contact

The forces acting during one-point contact are shown in

Figure 10-51 A derivation analogous to that for chamfer

crossing begins with the geometric constraint

and

10.J.3 Two-Point Contact

Whereas during chamfer crossing and one-point contact

we needed to find the forces before we could find U and 0, the reverse is true during two-point contact We find U and

and yields

FIGURE 10-51 Forces Acting During One-Point Contact.

and

where

10.J APPENDIX 287

0 via geometric compatibility

which reduces to Equation (10-2) when 9 is small The

relation between C7, UQ, 9, OQ, and eo during one-point

contact is obtained by combining Equation (10-15) and

Equation (10-23):

where

A force analysis based on the right side of

Fig-ure 10-18 may be used to determine when two-point

con-tact begins and ends The result, simplified for the case

where KQ ^> K XL2 and KgOo ^> iiK xe'§r is

for the termination of two-point contact and the start of

line contact The values of 0 at which these events occur

may be obtained by substituting and Equation (10-32) and

Equation (10-33) into Equation (10-2)

10.J.4 Insertion Forces

Insertion force during chamfer crossing is obtained by

substituting Equation (10-20) and Equation (10-21) and

into Equation (10-19) and Equation (10-20) to yield

Equa-and

where

and

If Equation (10-2) is substituted for 9 in

Equa-tion (10-29), we obtain the corresponding relaEqua-tion for

two-Doint contact:

Putting these into Equation (10-40) yieldsThese may be combined to yield

10.J.5 Computer Program

Table 10-2 contains the listing of a TRUE BASIC

com-puter program that calculates and plots all of the variables

discussed in this appendix This program provided the

"theory" lines in Figure 10-24 and Figure 10-25 The

fol-lowing is a brief discussion of how this program works and

how the variable names in it correspond to names used in

this chapter The code for this program is on the CD-ROM

that is packaged with this book as an "exe" that will run

on most PCs.

The first few lines express input data, which may be stored in the program or typed in by the user Such data include stiffnesses of the supports, clearance ratio between peg and hole, coefficient of friction, location of the sup- port compliance center, and the initial lateral and angular

errors Note that L e6 = 0 should not be used To simulate

small values for L g , one may use L g = /xr The program

TABLE 10-2 Listing of BASIC Program for Insertion Force

1000 REM PROGRAM FOR INSERTION FORCE

1010 REM BASED ON EQUATIONS IN THIS CHAPTER.

1020 REM THIS PROGRAM IS IN TRUE BASIC FOR THE MACINTOSH.

1030 REM VALUES OF COEFFICIENTS AND CONSTANTS ARE METRIC AND

1040 REM CORRESPOND TO EXPERIMENTAL DATA IN TABLE 10-1.

1140 INPUT PROMPT "TYPE SP FOR SCREEN PRINT, SG FOR SCREEN GRAPH":AN$

1150 IF (AN$ o "SP") AND (AN$ o "SG") THEN GOTO 1140

1160 INPUT PROMPT "INITIAL THETA ":TO

1170 IF AN$="SP" THEN PRINT "L FX FZ Ml Fl"

1270 LET L2 = (BE - SQR (BE ~ 2 - 4 * AL * GA)) / (2 * AL)

1280 LET L4 = (BE + SQR (BE A 2 - 4 * AL * GA)) / (2 * AL)

1290 LET LT = (4 * A + 2 * Cl * MU * D) * C * D

1300 LET L B = 2 * A * T O + B * ( 2 - M U * D / LG) + Cl * (TO * MU * D - C * D)

1310 LET LS = LT / LB

REM PLOT AXES

SET WINDOW -1,4*L2*SCALD + 2,-4,FM +10

* LL * EP / (KX * LG * LL + KT) TC

!•>-£0

#0

Lg D eo F G

£2 1'2 I*

F z max during two-point contact

C

e o /i (contact force) /2 (contact force)

F X ,F,,M

9 at end of chamfer crossing

9 when two point contact begins

Next is a short routine that plots axes on the screen.

The next few lines compute I* and F m , the depth at

which maximum insertion force occur and that force The

values of li and i' 2 where two-point contact begins and possibly ends are also computed here.

TYPE SP FOR SCREEN PRINT, SG FOR SCREEN GRAPH SG INITIAL THETA 002

FIGURE 10-52 Sample Output from BASIC Program for Insertion Force.

The main program loop is next, stepping through

val-ues of insertion depth t from zero to 4 times the predicted

li- The first part of this loop calculates insertion forces

during one-point contact When insertion depth exceeds

ti, the corresponding values for two-point contact are

calculated At the end of each pass, values are printed

and plotted

The last part of the program calculates two values

as-sociated with chamfer crossing: the value of 9 just at the

end, where one-point contact begins, and the insertion

force at that point One may assume that chamfer crossing

force and angle 9 each increase linearly during fer crossing, with force starting at zero and 9 starting

cham-at 0 0 Finally, there is a summary printout that repeats inputdata

-Correspondence of variable names is given inTable 10-3

Figure 10-52 is a sample of graphic output from thisprogram

ASSEMBLY OF COMPLIANT PARTS

"Mating one pin and socket isn't so hard Mating 100 at once can require

a hydraulic jack."

11.A INTRODUCTION

Chapter 10 dealt with compliantly supported rigid parts

entering rigid holes It was shown that insertion force and

two different failure modes depended on three basic

fac-tors: geometry, compliance, and friction The main design

parameter at our disposal was shown to be the location of

the compliance center of the rigid part's support

Different problems and design opportunities arise when

at least one mating part is compliant In particular, we shall

see in this chapter that the shape of at least one mating

sur-face can be varied so as to greatly affect the mating force.1

The first part of the chapter presents analytical models for

the main physical phenomena The second part develops

those models for several general cases The third part

fo-cuses specifically on opportunities for designing mating

surfaces, while the last part presents experimental

verifi-cations of the theory

Figure 11-1 exhibits numerous applications of

compli-ant parts, including electric connectors, door latches, snap

fits, and light bulb sockets Figure 11-2 shows two

sim-plified geometries that contain the elements considered in

this chapter Compliant sheet metal parts are not treated

in this chapter

11.A.1 Motivation

Compliant part mating is interesting both theoretically and

practically The theoretical issues are similar to those of

rigid parts in the sense that the same factors dominate

the mating behavior: geometry, compliance, and friction

'This chapter is based on Chapter 6 of [Nevins and Whitney].

However, because the parts are compliant in some places,

it is difficult to generate high enough forces to cause ing to occur At the same time, it is still possible to ob-serve phenomena similar to jamming Such events ariseduring chamfer crossing, when an entering part can be-come stuck against the chamfer If the parts are delicate,

wedg-as is the cwedg-ase with electrical connectors, the insertion forcecan build up to the point where the parts are damaged ordestroyed Theoretical models of compliant part matingcan be used as design guides to achieve desirable assem-bly features and avoid part damage and excessive matingforce

From a practical point of view, compliant part matingtypically involves substantial insertion force This forcecan act for good or ill, depending on the situation Sincemany compliant part mates are accomplished with barehands, the amount of force needed cannot be so high thatassembly becomes impossible This can happen with elec-trical connectors, especially if, as is common, twenty-five,fifty, or more pins must be mated to sockets simultane-ously In electrical connectors, this mating force arisesfrom the need to spread apart portions of the socket elasti-cally because a compressed socket is necessary in order toattain high enough contact force between pin and socket

to reduce the contact resistance and allow the connector

to function electrically

Situations like this arise in many compliant part matingsituations: Too large mating force will prevent assembly orcause damage, while too small mating force will preventthe item from functioning properly when assembled The-ory can come to the rescue here, permitting the engineer

to design the parts so that both needs can be met

293

FIGURE 11-1 Examples of Compliant Parts Shown

here are door latches, clamps, and electrical connectors The geometries look superficially different but all can be modeled mathematically, and the equations are similar in all the cases.

FIGURE 11-2 Models of Compliant Part Mating, (a) Rigid peg and compliant hole A

single compliant wall is shown, but both walls may be modeled as compliant if desired Both peg and hole mating surfaces are shown with shapes that may be represented mathemati- cally Different shapes give different insertion force behavior, (b) Rigid wall and compliant peg The peg is modeled as having two com- pliant sides, but one side may be modeled as rigid if desired The hole has a straight cham- fer shape while the peg's compliant elements are shown as lines that make a point contact with the hole The chamfer may be given a shape as shown in (a) if desired.

11.A INTRODUCTION 295

This chapter will explore two theoretical conditions:

a Rigid peg, compliant hole (Figure 1 l-2a)

b Compliant or compliantly held peg, rigid chamfered

hole (Figure 11-2b)

Before considering the theory, we will take a look at a

practical example

11 A.2 Example: Electrical Connectors

Figure 11-3 shows some typical shapes of electrical

con-nectors The pins may have a variety of nose shapes, with

spherical being the most common and easy to fabricate

However, as we shall see later in this chapter, tapered noses

can greatly reduce the insertion force Several socket

de-signs are also shown The compliant element is the

elec-trical contact spring The socket is manufactured in such

a way that the contact spring interferes with the pin as the

pin is inserted and deflects the contact spring elastically

This guarantees that there is a residual contact force when

assembly is finished This contact force ensures a good

electrical contact Connectors that carry large amounts of

current have thick material sections to reduce their

electri-cal resistance The contacts typielectri-cally behave as cantilever

beams, and their stiffness is governed by their thickness

and length Thus the contact springs in high-current

con-nectors tend to be quite stiff, and the insertion forces can

be large A critical design challenge is to achieve high tact force while avoiding high insertion force Since thesame element, the contact spring, is responsible for bothphenomena, this goal would appear to be out of reach Infact, however, it is largely achievable, and we will show

con-in this chapter how to address it

Figure 11-4 illustrates compliant part mating eventsfor some of the connector pins and sockets shown in Fig-ure 11-3 The insertion force comes from the axial com-ponents of contact and friction forces The friction force

is proportional to the contact force, which in turn is portional to the lateral stiffness and deflection of the con-tact spring The amount of insertion force generated alsodepends on the coefficient of friction and the angle ofthe surfaces at the pin-spring contact point Figure 11-4also illustrates a design in which the insertion processdoes not go as desired but instead buckles the contactspring

pro-The remainder of this chapter is devoted to modelingthe geometry and force characteristics of parts like these

so that their shapes can be designed to achieve desiredinsertion force and contact force behavior

Significantly more complex conditions have beensolved: compliant peg/compliant hole and minimum en-ergy chamfers in [Hennessey] and a three-dimensionalpart mating theory in [Gustavson] The interested readershould consult these references

FIGURE 11-3 Examples of Real Compliant Parts.

These are schematic drawings of electrical pins and sockets Each will display very different insertion force versus insertion distance behavior Some versions of shapes 1, 3, and 4 run the risk of the pin jamming on the chamfer of the compliant element in the socket Real connector pins have diameters ranging from 1 to 4 mm and lengths ranging from 4 to 10 mm.

FIGURE 11-4 Schematic Diagram of Insertion Motions and Forces in Electrical Connectors Left: When the pin

touches the flexible spring inside the socket, a contact force and a friction force arise The axial components of these forces

are felt as insertion force Center: In a normal successful compliant pin-socket assembly, the pin deflects the spring to one

side and enters the socket The spring is compressed in the final assembled state, giving rise to a contact force that provides

firm electrical contact Right: The pin has jammed against the contact spring because the contact force was inside the friction

cone The spring buckled under this axial load and the socket has been destroyed.

11.B DESIGN CRITERIA AND CONSIDERATIONS

11.B.1 Design Considerations

Compliant parts are designed to perform various

func-tions in various environments The parts may be mated by

hand or machine They may be delicate or rugged The

de-signer may, for example, wish easy insertion and difficult

withdrawal, or may wish to signal incomplete mating by

having the parts pop apart There are so many criteria that

we list only a few, involving the insertion force (force in

the direction of insertion) or withdrawal force

1 Avoid sharp discontinuities in force versus insertion

depth

2 Minimize mechanical work during insertion

3 Minimize the peak value attained by the insertion

force during insertion

4 Achieve a specific pattern of force versus depth

5 Achieve a specific ratio of insertion force to

with-drawal force

A number of design features influence insertion force:

1 Peg nose shape

2 Number of springs (compliant members) making up

the compliant hole

3 Entry shape of the spring

4 Speed of entry (quasi-static or dynamic)

5 Type of spring deflection (linear, nonlinear)

6 Spring preload

7 Rigid, compliant, or compliantly held rigid pegs

8 Straight or tilted initial entry of the peg into the holeThe most influential feature is the shape of the contact-ing surfaces These surfaces are typically the tips of pegsand the mouths of holes

Four basic types of insertion force behavior have beenidentified Each corresponds to a particular type of matingsurface shape The shapes could be on either the peg orthe hole

a Linear shape (Figure 11-5a)—This is the mostcommon and provides linear force versus depthbehavior The maximum force occurs at the end ofinsertion and could be very high

b Convex shape (Figure 1 l-5b)—Making the surfaceconvex allows shaping of the force versus depthcurve Various geometrical forms have been triedincluding circular arcs, parabolas and logarithmiccurves

11.B DESIGN CRITERIA AND CONSIDERATIONS 297

11.B.2 Assumptions

FIGURE 11-5 Insertion Force Versus Insertion Depth for

Four Generic Chamfer Shapes.

c Constant force shape (Figure 1 l-5c)—A particular

convex shape with a complex descriptive equation

can produce constant insertion force throughout the

insertion This behavior results in minimum peak

force

d Concave shape (Figure 11-5d)—Reversing the arc

provides less force for beginning depths but very

large forces near the end

Table 11-1 lists the assumptions used in this chapter Thecomments indicate extensions that might be added

11 B.3 General Force Considerations

While a significant variety of mathematical models forcompliantly held peg into hole or peg into compliant holecan be created, the complexity lies in describing the orien-tation geometry and the elastic behavior of the compliantelements If we focus on the peg/compliant member inter-face at any instant during part mating (see Figure 11-6),

we may write the following basic equations for the

inser-tion and lateral forces F z and F x (subscript / = insertion,

W = withdrawal) acting on the peg during insertion and withdrawal in terms of the normal contact force F N andinterface angle 0

Ratios of F z to F x versus </> are plotted in Figure 11-7for the case of insertion and in Figure 11-8 for the case ofwithdrawal These figures show that the ratio of insertionforce to lateral force during insertion is larger for largercoefficient of friction and smaller for larger interface an-gle During withdrawal, the ratio is again smaller for largerangle but smaller for larger friction coefficient Note thatfor straight entry shapes, the angle is constant, whereasfor curved shapes the angle changes during insertion.These figures show that three factors control the matingforces of compliant peg-hole combinations:

1 The normal (or contact) force

2 The slope at the interface point where 0 =tan"1 (slope)

3 The friction coefficient /JL

Establishing and controlling these three factors is damental to compliant peg-hole design They are defined

fun-as follows:

11.B.3.a The Normal Force

The normal force, which produces the insertion force,

as given by Equation (11-1), is created by deflection ofthe compliant member(s) Certain peg-hole combinations

TABLE 11-1 Assumptions for Analysis of Compliant Part Mating

1 Two-dimensional cases only.

2 Peg travels along the centerline of hole.

3 Peg has prescribed lateral position.

a For one compliance—one rigid wall case: peg rides along wall

opposite compliance (see Figure 1 l-9a and Figure 1 l-12a).

b For two equal symmetrical compliance case: peg and "hole"

centerlines are coincident (see Figure 1 l-9b and Figure 1 l-12b).

4 Compliant elements are an integral part of the mounting; friction

coefficient is uniform on all contacting surfaces.

5 Deflection of the compliance(s) is rigid body motion with respect

to a single point, with compliance concentrated at that point.

6 Spring has no preload at initial contact with peg.

7 Conditions are quasi-static; motion does not create need for dynamic

considerations.

1 Third dimension could be added.

2 Lateral and angular misalignments can be added as can inclined approach paths.

3 Alternately:

a Peg-wall contact may not occur at all.

b Compliance may not be shape-symmetric or have equal stiffnesses Only one compliance may be contacted by peg.

4 Compliant elements could be separate parts made of different materials requiring specification of two or more friction coefficients.

5 Small deflection beam theory or large deflection theory can be added.

6 Spring preload can be added.

7 Dynamics may play a role in compliant part mating.

FIGURE 11-6 Definition of Forces and Directions During

Insertion and Withdrawal in Compliant Part Mating

Dur-ing insertion, the contact force generates the friction force

/iF/v, on the compliant member, whereas during withdrawal

it generates the friction force ^FN W

-contain a spring whose action can be analyzed only by

large-deflection (nonlinear) theory; they are not included

here For "small" deflections, two types of behavior are

possible:

1 Elastic deflection

2 Rigid body motion with respect to a single point

where all the compliance is concentrated

This chapter analyzes only the second type; the

mathe-matics is considerably less complex while agreement with

experimental results is good

FIGURE 11-8 Ratio of Insertion Force to Lateral Force During Withdrawal Based on Equation (11-1).

11 B.S.b Slope at the Peg/Spring Interface

The normal force at the point of contact is oriented alongthe line joining the center of curvature of the peg's surface

FIGURE 11-7 Ratio of Insertion Force to Lateral Force During Insertion Based on Equation (11-1) As the coeffi-

cient of friction IJL increases, the ratio of insertion force to

lat-eral force rises The ratio falls as the interface angle 0 rises.